by the definition of exactness, the Fermat rule (see for example [1]}) implies, in turn, that \((\bar{y}, 0) \in \partial f(\bar{u},x^\star )\)  provided that \(f(\bar{u}, x^\star )\in \mathbb {R}\) . The latter holds because \(\varphi \)  is proper. Thus, \(\bar{u}\)  furnishes \(u^\star \)  in the assertion.\(\hrule height1pt \hbox{\vrule width1pt height5pt\hspace{5.0pt}\vrule width1pt} \hrule height1pt\)\(\)
